🤖 AI Summary
This work addresses the computational inefficiency of traditional kernel Stein discrepancy (KSD) tests, which suffer from quadratic time complexity due to their reliance on U- or V-statistics and require computationally intensive bootstrap procedures to approximate the null distribution. To overcome these limitations, the authors propose an accelerated KSD test based on the Nyström approximation. They provide the first theoretical guarantee that this approach preserves asymptotic type-I error control and local consistency within a bootstrap framework, while substantially reducing computational cost. Empirical evaluations on spherical and functional data demonstrate that the accelerated method achieves statistical performance comparable to the original KSD test but with significantly improved computational efficiency, thereby enabling scalable and theoretically sound nonparametric goodness-of-fit testing.
📝 Abstract
Kernel Stein discrepancy (KSD) is among the most popular goodness-of-fit (GoF) measures on general domains with a large number of successful deployments. One of the main applications of KSD is in constructing powerful GoF tests. However, tests relying on the classical U-/V-statistic-based KSD estimators have two major drawbacks. (i) Their runtime scales quadratically in the number of samples. (ii) Their asymptotic null distribution is computationally intractable in most cases, typically handled by bootstrapping. While it is known that the Nyström method permits accelerating KSD estimation with no loss of statistical accuracy under mild conditions, to the best of our knowledge, the fundamental question of its impact on bootstrap-based GoF testing is open; resolving this question is the focus of the current paper. In particular, we prove that the key properties of the quadratic-time bootstrapped KSD-based GoF test (asymptotic level and local consistency) are preserved by its Nyström acceleration. We numerically demonstrate the efficiency of the accelerated KSD estimator and bootstrap in the context of GoF testing of spherical and functional data. Our numerical results show that the Nyström-accelerated method performs statistically on-par with the quadratic-time approach, while requiring substantially smaller runtime.